Amoxicillin-induced rash: This is a common adverse reaction to amoxicillin, especially in children. The rash typically appears 3-7 days after starting the antibiotic, which fits the timeline in this case.

Viral exanthem: The child had a recent history of cough and fever, which could be indicative of a viral infection. Many viral infections can cause a rash, and the timing could be coincidental with the amoxicillin treatment.
Allergic intertrigo or contact dermatitis: The rash could be an allergic reaction to another substance, such as a soap or lotion, especially if it's localized to the trunk.

Kawasaki disease: Although less common, Kawasaki disease is a serious condition that can present with fever, rash, and other symptoms. It's crucial to consider this diagnosis, especially if the child has other systemic symptoms like conjunctivitis or swollen lymph nodes.
Bacterial sepsis or meningitis: If the child's otitis media was not fully treated or if there's a complication, bacterial sepsis or meningitis could be a life-threatening condition that needs immediate attention. The presence of a rash could be a sign of disseminated infection.
Stevens-Johnson syndrome: This is a rare but serious condition that can be triggered by medications, including antibiotics like amoxicillin. It's characterized by a severe skin and mucous membrane reaction.

Juvenile idiopathic arthritis (JIA): Some forms of JIA can present with a rash, although this would be unusual in a 7-month-old child.
Infectious diseases like scarlet fever or measles: Although vaccination has made these diseases less common, they could still be considered in the differential diagnosis, especially if the child is not up to date on vaccinations or has been exposed to someone with the disease.